Which properties of a random sequence are dynamically sensitive? (English) Zbl 1021.60055

Benjamini, Itai; Häggström, Olle; Peres, Yuval; Steif, Jeffrey E.

Ann. Probab. 31, No. 1, 1-34 (2003).

A lot of classical results in probability theory, for instance the strong law of large numbers, the law of iterated logarithm, and so on, concern almost-sure properties of sequences \(\{X_n\}\) of i.i.d. random variables.
Consider a sequence of i.i.d. random variables \(\{X_n\}\) with state space \(S\) and common law \(\nu\), where each variable is replaced independently by an independent variable with the same law according to a Poisson clock, i.e., for each \(n\in\mathbb{N},\) let \(X(t):=\{X_n(t)\},\) \(t\geq 0\), be an independent process which at rate 1 replaces its current value by an independent sample from \(\nu.\) The distribution of \(X(t)\) is \(\mu=\nu^{\mathbb{N}}\) for every \(t\geq 0,\) thus any Borel set \(A\subset S^{\mathbb{N}}\) such that \(\mu(A)=1\) satisfies for all \(t\geq 0,\) \(P\{X(t)\in A\}=1\) and \(P\{X(t)\in A\text{ for Lebesgue-a.e. }t\}=1.\) The almost-sure properties of \(\{X_n\}\), i.e., the subsets \(A\) of \(S^{\mathbb{N}}\) with \(\mu(A)=1\) are classified by the authors as (dynamically) stable or sensitive according to whether or not the property: for all \(t\geq 0,\) \(P\{X(t)\in A\text{ for Lebesgue-a.e. }t\}=1\) can be strengthened to \(P\{X(t)\in A\text{ for all }t\}=1\).
The authors prove that the law of large numbers and the law of iterated logarithm are dynamically stable while run tests are dynamically sensitive. They obtain multi-fractal analysis of exceptional times for run lengths and for prediction.
For the simple random walk in \(\mathbb{Z}^d\) the authors prove that the transience is dynamically stable for \(d\geq 5,\) and dynamically sensitive for \(d=3,4.\) Moreover, for \(d=3,4\) the nonempty random set of exceptional times \(t\) where the random walk is recurrent has Hausdorff dimension \((4-d)/ 2\) a.s.

Reviewer: Roman Urban (Wrocław)

MSC:

60J25	Continuous-time Markov processes on general state spaces
28A80	Fractals
60F15	Strong limit theorems
28A78	Hausdorff and packing measures

Keywords:

dynamical limit theorems; exceptional times; run tests; von Mises-Church randomness; random walk; Hausdorff dimension

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